By Iain Adamson

This publication has been referred to as a Workbook to make it transparent from the beginning that it isn't a standard textbook. traditional textbooks continue through giving in each one part or bankruptcy first the definitions of the phrases for use, the techniques they're to paintings with, then a few theorems concerning those phrases (complete with proofs) and eventually a few examples and workouts to check the readers' figuring out of the definitions and the theorems. Readers of this e-book will certainly locate the entire traditional constituents--definitions, theorems, proofs, examples and exercises yet no longer within the traditional association. within the first a part of the e-book can be came upon a brief overview of the elemental definitions of common topology interspersed with a wide num ber of workouts, a few of that are additionally defined as theorems. (The use of the observe Theorem isn't really meant as a sign of trouble yet of significance and value. ) The routines are intentionally no longer "graded"-after all of the difficulties we meet in mathematical "real life" don't are available order of trouble; a few of them are extremely simple illustrative examples; others are within the nature of educational difficulties for a conven tional path, whereas others are relatively tough effects. No options of the workouts, no proofs of the theorems are integrated within the first a part of the book-this is a Workbook and readers are invited to aim their hand at fixing the issues and proving the theorems for themselves.

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Cor. 22. Let n-+ X; i ~ ai be a finite sequence on X, an ordered D field, such that ~ al = 0. Then, for all i En, ai = 0. ien Prop. 23. An ordered ring is without divisors of zero. D Absolute value Let X be an ordered ring. Then a map X-+ X; x ~1 xI is defined by setting I 0 I = 0, I x I = x, if x > 0, and I x I = -x if -x > 0. The 34 REAL AND COMPLEX NUMBERS element I x I is said to be the absolute value of x and, for any a, b I b - a I is said to be the absolute difference of a and b. Prop. 24.

Since each real number is, by Prop. 59, the limit of a convergent sequence of rational numbers and since, by Cor. 52, the limit of a convergent sequence on R is unique, it follows that f sends each element of R to itself. D When b = 2, a sequence of the type constructed in Prop. 59 is said to be a binary expansion for the real number, and when b = 10 the sequence is said to be a decimal expansion for the real number. 61. Discuss to what extent the binary and the decimal expansions for a given real number are unique.

That is, for any z, z' E C, z + z' =; + ;', zz' =-:;. ;;, T = 1 and, for any z :;:z: 0, z- 1 = (z)- 1• Also, for any z E C, ('i) = z; z + z and i(z - z) are 47 THE COMPLEX FIELD real numbers and z z is a non-negative real number, if, and only if, z is real. D + z being equal to z + For any z = x iy E C, with x, y E R, x = i(z z) is said to be the real part of :<1 andy = ti(z - z) is said to be the pure imaginary part of z. The real part ofzwill be denoted by rezand the pure imaginary part of z will be denoted by pu z (the letters im being reserved as an abbreviation for 'image').